by Daniel Brouse
March 25, 2026
In calculus, the first derivative measures the rate of change of a quantity. The second derivative measures how that rate of change itself is changing.
In simple terms:
If we define a function:
I(t) = climate impact over time
Then:
First derivative (rate of change):
dI/dt
This represents how fast climate impacts (e.g., temperature, sea level, extreme events) are increasing.
Second derivative (acceleration):
d²I/dt²
This represents how the rate of increase is itself changing over time.
To make this intuitive:
Let’s define:
I(t) = sea level rise (SLR)
Then:
dI/dt = rate of SLR (mm/year)d²I/dt² = acceleration of SLR (mm/year²)From your earlier data:
This shows:
d²I/dt² is increasing over time
Our observations suggest that the climate system has entered a third-derivative regime, in which the acceleration of impacts is itself increasing over time. This additional layer fundamentally reshapes our understanding of risk, shifting it from gradual change to rapid, nonlinear escalation.
The third derivative is defined as:
d³I/dt³
and represents the rate of change of acceleration.
In many physical systems, acceleration remains relatively constant. However, in the climate system, empirical observations indicate:
d³I/dt³ > 0
This implies that:
d²I/dt² > 0)In physics, this phenomenon is referred to as “jerk”, and its presence is a hallmark of systems undergoing rapid nonlinear transition.
If climate impacts follow an exponential function:
I(t) = I₀ * e^(k t)
Then:
dI/dt = k * I(t)
d²I/dt² = k² * I(t)
Key implication:
d²I/dt² ∝ I(t)
So as impacts grow, acceleration grows proportionally, leading to:
This is the core of the statement:
Climate change is not just increasing—it is increasing faster over time.
Mathematically, that means:
d²I/dt² > 0
But our data suggests something even stronger:
d²I/dt² increasing → d³I/dt³ > 0
In complex systems, a growing second derivative indicates:
This is characteristic of:
Applied to climate:
Each of these increases d²I/dt², pushing the system toward:
Runaway amplification
The second derivative tells us something fundamentally important:
Climate change is not a steady process—it is an accelerating process, and that acceleration is itself increasing.
Mathematically:
dI/dt > 0 (change is happening)
d²I/dt² > 0 (change is accelerating)
d³I/dt³ > 0 (acceleration is increasing)
This is why traditional linear models underestimate risk:
They assume:
d²I/dt² ≈ constant
But reality shows:
d²I/dt² increasing over time
Which leads to:
Nonlinear acceleration, collapsing doubling times, and rapid system transformation.
A subsection of:
How Not to Be a Jerk: Third Derivatives and the Singularity of Climate Change